Tijms approximations

Abstract

A compound distribution is in general complex and hence a closed form is not available for its tail $\bar G(x)$ except for a few special cases. In the previous chapters, we have discussed upper and lower bounds on compound distribution tails. In this chapter various approximations are considered. These approximations are motivated by an approximation given in Tijms (1986, p. 61), in which Tijms proposed the use of a combination of two exponentials to approximate a compound geometric tail. The parameters are chosen as such that the probability mass at 0 and the mean of two distributions are matched. We extend this idea by considering a combination of an exponential distribution with the adjustment coefficient as its parameter and a general distribution. Again, the probability mass at 0 and the mean are matched. As will be demonstrated, this approximation provides a very satisfactory result when the number of claims distribution is asymptotically geometric. For a large class of individual claim amount distributions, the approximating distribution has the same asymptotic behaviour as that of the compound distribution.